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Given a closed orientable surface (\Sigma) of genus at least two, weestablish an affine isomorphism between the convex compact set ofisotopy-invariant topological measures on (\Sigma) and the convex compact setof additive functions on the set of isotopy classes of certain subsurfaces of(\Sigma). We then construct such additive functions, and thus isotopy-invarianttopological measures, from probability measures on (\Sigma) together with someadditional data. The map associating topological measures to probabilitymeasures is affine and continuous. Certain Dirac measures map to simpletopological measures, while the topological measures due to Py and Rosenbergarise from the normalized Euler characteristic.

Isotopy-invariant topological measures on closed orientable surfaces of higher genus